行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}{-}1 & 1 & 2 & {-}2 & {-}2 \\ 2 & {-}1 & 2 & 2 & {-}1 \\ 0 & 0 & {-}1 & 2 & 0 \\ 2 & {-}1 & {-}2 & {-}2 & {-}1 \\ 2 & {-}1 & {-}2 & 0 & {-}1\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 & {-}2 \\ 2 & {-}1 & 2 & 2 & {-}1 \\ 0 & 0 & {-}1 & 2 & 0 \\ 2 & {-}1 & {-}2 & {-}2 & {-}1 \\ 2 & {-}1 & {-}2 & 0 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第5列で展開する : [-2, -1, 0, -1, -1]

\( \qquad\quad = (-1)^{5-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| + (-1)^{5-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| + (-1)^{5-4}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| + (-1)^{5-5}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{5-1}\times(-2)\times0 + (-1)^{5-2}\times(-1)\times(-2) + (-1)^{5-4}\times(-1)\times(-10) + (-1)^{5-5}\times(-1)\times(-12) \)

\( \qquad\quad = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [2, 2, -2, 0]

\( \qquad\quad = (-1)^{4-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}1 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times2\times0 + (-1)^{4-2}\times2\times0 + (-1)^{4-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}1 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, -2, -2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times0 + (-1)^{3-2}\times(-2)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -2, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-2)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| = 2\times0 - (-1)\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, -2, 0]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}1 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times0 + (-1)^{4-2}\times2\times0 + (-1)^{4-3}\times(-2)\times(-1) \)

\( \qquad\quad = -2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}0 & 0 & {-}1 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-1, -2, -2]

\( \qquad\quad = (-1)^{3-1}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-1)\times0 + (-1)^{3-2}\times(-2)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & {-}2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -2, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-2)\times(-1) + (-1)^{3-3}\times(-2)\times(-1) \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times(-1) + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| = (-1)\times0 - 1\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & 0\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, 2, 0]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times0 + (-1)^{4-2}\times2\times(-1) + (-1)^{4-3}\times2\times4 \)

\( \qquad\quad = -10 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| = 2\times0 - (-1)\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times(-1) + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| = (-1)\times0 - 1\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-2)\times(-1) \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 & {-}2 \\ 2 & {-}1 & 2 & 2 \\ 0 & 0 & {-}1 & 2 \\ 2 & {-}1 & {-}2 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [-2, 2, 2, -2]

\( \qquad\quad = (-1)^{4-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 2 & {-}1 & {-}2\end{array}\,\right| + (-1)^{4-4}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 0 & 0 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times(-2)\times0 + (-1)^{4-2}\times2\times(-1) + (-1)^{4-3}\times2\times4 + (-1)^{4-4}\times(-2)\times1 \)

\( \qquad\quad = -12 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & {-}1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times0 + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| = 2\times0 - (-1)\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 0 & 0 & {-}1 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, -1, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times(-1)\times(-1) + (-1)^{3-3}\times(-2)\times0 \)

\( \qquad\quad = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 0 \\ 2 & {-}1\end{array}\,\right| = 0\times(-1) - 0\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| = (-1)\times0 - 1\times0 = 0 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 2 & {-}1 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, -2]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| + (-1)^{3-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times2\times(-1) + (-1)^{3-3}\times(-2)\times(-1) \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 2 & {-}1\end{array}\,\right| = 2\times(-1) - (-1)\times2 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 1 & 2 \\ 2 & {-}1 & 2 \\ 0 & 0 & {-}1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [2, 2, -1]

\( \qquad\quad = (-1)^{3-1}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| + (-1)^{3-2}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| + (-1)^{3-3}\times(-1)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times2\times0 + (-1)^{3-2}\times2\times0 + (-1)^{3-3}\times(-1)\times(-1) \)

\( \qquad\quad = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & {-}1 \\ 0 & 0\end{array}\,\right| = 2\times0 - (-1)\times0 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 0 & 0\end{array}\,\right| = (-1)\times0 - 1\times0 = 0 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 1 \\ 2 & {-}1\end{array}\,\right| = (-1)\times(-1) - 1\times2 = -1 \)